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A partially ordered set (poset for short) is an ordered pair = (,) consisting of a set (called the ground set of ) and a partial order on . When the meaning is clear from context and there is no ambiguity about the partial order, the set X {\displaystyle X} itself is sometimes called a poset.
A given partially ordered set may have several different completions. For instance, one completion of any partially ordered set S is the set of its downwardly closed subsets ordered by inclusion. S is embedded in this (complete) lattice by mapping each element x to the lower set of elements that are less than or equal to x.
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset). The most familiar example is the completeness of the real numbers. A special use of the term refers to complete partial orders or complete lattices. However, many other interesting notions ...
If the preordered set (,) also happens to be a partially ordered set (or more generally, if the restriction (,) is a partially ordered set) then is a maximal element of if and only if contains no element strictly greater than ; explicitly, this means that there does not exist any element such that and .
In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by P op or P d.This dual order P op is defined to be the same set, but with the inverse order, i.e. x ≤ y holds in P op if and only if y ≤ x holds in P.
A subset F of a partially ordered set (P, ≤) is a filter or dual ideal if the following are satisfied: [3] Nontriviality The set F is non-empty. Downward directed For every x, y ∈ F, there is some z ∈ F such that z ≤ x and z ≤ y. Upward closure For every x ∈ F and p ∈ P, the condition x ≤ p implies p ∈ F.
The set of all linearly independent subsets of a vector space V, ordered by inclusion. The set of all partial choice functions on a collection of non-empty sets, ordered by restriction. The set of all prime ideals of a ring, ordered by inclusion. The specialization order of any sober space is a dcpo.
If is an arbitrary set, its power set (), ordered by set inclusion, is always a Boolean algebra and hence a poset, and ultrafilters on () are usually called ultrafilters on the set. [ note 1 ] An ultrafilter on a set X {\displaystyle X} may be considered as a finitely additive 0-1-valued measure on P ( X ) {\displaystyle {\mathcal {P}}(X)} .